I noticed that the compositing is applied after the image is rendered. Does this mean that when working with an 8-bit Output that the compositor can affect image quality negatively? (i.e. artifacts like banding etc.)
No, the compositor will not degrade rendered images. The compositor works with 32bit float precision, in scene referred linear (meaning that there is way more information that what can be displayed in the monitor).
Of course, there could be intentional or unintentional degrading in the compositor caused by incorrectly connected nodes and/or poor judgement by the user.
Banding would happen when the image is saved to 8 bit formats, as this method of encoding cannot resolve subtle gradation with only 255 levels for each color. Similar colors, and dark, almost monochromatic gradients, will suffer greatly.
To disguise banding in the rendered image, use dithering.
Note that even if the image is saved as 16bit per channel, it is possible that it cannot be displayed at a better quality. Most monitors display only 8 bits per channel (and some even less than that). Banding then is inevitable (just pause any of your favorite movies during streaming and you will see what I'm talking about). Banding might disappear if the higher bit depth images are displayed properly in a 10 bit monitor or better (retina display or any professional grade monitor or video projector)